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Optimal control of the Dyson equation and large deviations for Hermitian random matrices
Abstract
Using novel arguments as well as techniques developed over the last twenty years to study mean field games, in this paper (i) we investigate the optimal control of the Dyson equation, which is the mean field equation for the so-called Dyson Brownian motion, that is, the stochastic particle system satisfied by the eigenvalues of large random matrices, (ii) we establish the well-posedness of the resulting infinite dimensional Hamilton-Jacobi equation,
(iii) we provide a complete and direct proof for the large deviations for the spectrum of large random matrices, and (iv) we study the asymptotic behavior of the transition probabilities of the Dyson Brownian motion. Joint work with Charles Bertucci and Pierre-Louis Lions.
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Prelims Corner is back for Michaelmas 2025!
We will be running Prelims Corner from 11am-12pm on Mondays outside L1 from Weeks 2-8. This is the perfect opportunity to receive support from an experienced DPhil student about particularly tricky maths problems, or just to meet and work with other Prelims students. Registration isn't required, so feel free to drop in and out as you need. Free snacks are provided!